Publications of (and about) Paul Erdös
Autor: Burr, Stefan A.; Erdös, Paul
Title: Completeness properties of perturbed sequences. (In English)
Source: J. Number Theory 13, 446-455 (1981).
Review: For each sequence S of positive integers, denote by P(S) the set of integers representable as a sum of distinct terms of S. The sequence S is called complete of P(S) contains all large integers, entirely complete if P(S) contains all positive integers, strongly complete if S remains complete after the removal of a finite number of terms and subcomplete if P(S) contains and infinite arithmetic progression (the definition on p. 446 has ``any'' instead of ``an''). Completeness is a delicate property in the sence that it can be destroyed by the removal of a few terms of the sequence. It is shown that even the less restrictive property of subcompleteness is not very robust since any sequence can be pertubed by adding integers (possibly \leq 0) of moderate modulus to each term to yield a sequence satisfying rather mild growth conditions has a slight pertubation wich is strongly complete. On the other hand it is shown that sufficiently rapidly growing sequences are not complete and that certain classes of pertubations of particular sequences are not complete. A number of open questions arising from this work are also discussed.
Classif.: * 11B99 Sequences and sets
11B25 Arithmetic progressions
11A99 Elementary number theory
Keywords: sequence of positive integers; sum of distinct terms; completeness; subcompleteness; incompleteness
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